## Analysis & PDE

• Anal. PDE
• Volume 10, Number 2 (2017), 323-350.

### Operators of subprincipal type

Nils Dencker

#### Abstract

In this paper we consider the solvability of pseudodifferential operators when the principal symbol vanishes of at least second order at a nonradial involutive manifold $Σ2$. We shall assume that the subprincipal symbol is of principal type with Hamilton vector field tangent to $Σ2$ at the characteristics, but transversal to the symplectic leaves of $Σ2$. We shall also assume that the subprincipal symbol is essentially constant on the leaves of $Σ2$ and does not satisfying the Nirenberg–Trèves condition ($Ψ$) on  $Σ2$. In the case when the sign change is of infinite order, we also need a condition on the rate of vanishing of both the Hessian of the principal symbol and the complex part of the gradient of the subprincipal symbol compared with the subprincipal symbol. Under these conditions, we prove that $P$ is not solvable.

#### Article information

Source
Anal. PDE, Volume 10, Number 2 (2017), 323-350.

Dates
Revised: 18 September 2016
Accepted: 1 November 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843424

Digital Object Identifier
doi:10.2140/apde.2017.10.323

Mathematical Reviews number (MathSciNet)
MR3619872

Zentralblatt MATH identifier
06693359

#### Citation

Dencker, Nils. Operators of subprincipal type. Anal. PDE 10 (2017), no. 2, 323--350. doi:10.2140/apde.2017.10.323. https://projecteuclid.org/euclid.apde/1510843424

#### References

• F. Cardoso and F. Trèves, “A necessary condition of local solvability for pseudo-differential equations with double characteristics”, Ann. Inst. Fourier $($Grenoble$)$ 24:1 (1974), 225–292.
• F. Colombini, L. Pernazza, and F. Trèves, “Solvability and nonsolvability of second-order evolution equations”, pp. 111–120 in Hyperbolic problems and related topics, edited by F. Colombini and T. Nishitani, International Press, Somerville, MA, 2003.
• F. Colombini, P. D. Cordaro, and L. Pernazza, “Local solvability for a class of evolution equations”, J. Funct. Anal. 258:10 (2010), 3469–3491.
• N. Dencker, “The resolution of the Nirenberg–Treves conjecture”, Ann. of Math. $(2)$ 163:2 (2006), 405–444.
• N. Dencker, “Solvability and limit bicharacteristics”, J. Pseudo-Differ. Oper. Appl. 7:3 (2016), 295–320.
• J. V. Egorov, “Solvability conditions for equations with double characteristics”, Dokl. Akad. Nauk SSSR 234:2 (1977), 280–282. In Russian; translated in Sov. Math. Dokl. 18 (1977), 632–635.
• A. Gilioli and F. Trèves, “An example in the solvability theory of linear PDE's”, Amer. J. Math. 96 (1974), 367–385.
• R. Goldman, “A necessary condition for the local solvability of a pseudodifferential equation having multiple characteristics”, J. Differential Equations 19:1 (1975), 176–200.
• L. H örmander, “The Cauchy problem for differential equations with double characteristics”, J. Analyse Math. 32 (1977), 118–196.
• L. H örmander, “Pseudodifferential operators of principal type”, pp. 69–96 in Singularities in boundary value problems (Maratea, Italy, 1980), edited by H. G. Garnir, NATO Adv. Study Inst. Ser. C: Math. Phys. Sci. 65, Reidel, Dordrecht, 1981.
• L. H örmander, The analysis of linear partial differential operators, I: Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften 256, Springer, Berlin, 1983.
• L. H örmander, The analysis of linear partial differential operators, III: Pseudodifferential operators, Grundlehren der Mathematischen Wissenschaften 274, Springer, Berlin, 1985.
• L. H örmander, The analysis of linear partial differential operators, IV: Fourier integral operators, Grundlehren der Mathematischen Wissenschaften 275, Springer, Berlin, 1985.
• G. A. Mendoza, “A necessary condition for solvability for a class of operators with involutive double characteristics”, pp. 193–197 in Microlocal analysis (Boulder, CO, 1983), Contemp. Math. 27, Amer. Math. Soc., Providence, RI, 1984.
• G. A. Mendoza and G. A. Uhlmann, “A necessary condition for local solvability for a class of operators with double characteristics”, J. Funct. Anal. 52:2 (1983), 252–256.
• G. A. Mendoza and G. A. Uhlmann, “A sufficient condition for local solvability for a class of operators with double characteristics”, Amer. J. Math. 106:1 (1984), 187–217.
• T. Nishitani, “Effectively hyperbolic Cauchy problem”, pp. 363–449 in Phase space analysis of partial differential equations, vol. 2, Scuola Norm. Sup., Pisa, 2004.
• P. R. Popivanov, “The local solvability of a certain class of pseudodifferential equations with double characteristics”, C. R. Acad. Bulgare Sci. 27 (1974), 607–609. In Russian.
• F. Trèves, “Concatenations of second-order evolution equations applied to local solvability and hypoellipticity”, Comm. Pure Appl. Math. 26 (1973), 201–250.
• F. Trèves, Introduction to pseudodifferential and Fourier integral operators, vol. 2: Fourier integral operators, Plenum Press, New York, 1980.
• P. R. Wenston, “A necessary condition for the local solvability of the operator $P\sb{m}\sp{2}(x,D)+P\sb{2m-1}(x,D)$”, J. Differential Equations 25:1 (1977), 90–95.
• P. R. Wenston, “A local solvability result for operators with characteristics having odd order multiplicity”, J. Differential Equations 28:3 (1978), 369–380.
• J. Wittsten, “On some microlocal properties of the range of a pseudodifferential operator of principal type”, Anal. PDE 5:3 (2012), 423–474.
• A. Yamasaki, “On a necessary condition for the local solvability of pseudodifferential operators with double characteristics”, Comm. Partial Differential Equations 5:3 (1980), 209–224.
• A. Yamasaki, “On the local solvability of $\smash{D\sp{2}\sb{1}}+A(x\sb{2},\,D\sb{2})$”, Math. Japon. 28:4 (1983), 479–485.